The time series of the returns for the 6 portfolios are plotted in Figure. \ref{fig:vwret_ts}. We notice that they all look stationary. By setting the y-axis with the same scale, we can see that SMLO and BIHI have larger spikes than other portfolios. Figure. \ref{fig:vwret_acf} shows non-significant autocorrelations for all 6 portfolios. Hence, we don't need to transform or difference the time series to make it stationary.
% it actually looks to me like a couple of the ACFs contain significance. What to do about that?

\begin{figure}[hbtp]
\centering
\includegraphics[width=7.5cm]{../results/vwret_ts.png}
\caption{The returns for the 6 portfolios.}
\label{fig:vwret_ts}
\end{figure}

We decomposition the returns into the trend(middle panel), the seasonal component(top panel), and residuals (bottom panel). Please see one example for SMLO shown in Figure. \ref{fig:smlo_stl}. From the trend, we cannot see a clear pattern. However, there is a yearly seasonal change of the returns. Hence, we fit an ARMA(p,q) model to the time series.

\begin{figure*}[tbp]
\centering
\subfigure[Autocorrelations of the 6 returns series] {
	\includegraphics[width=7.5cm]{../results/vwret_acf.png}
	\label{fig:vwret_acf}
}
\subfigure[The decomposition of returns for SMLO.] {
	\includegraphics[width=7.5cm]{../results/smlo_stl.png}
	\label{fig:smlo_stl}
}
\end{figure*}

The implemented arma.model.selection function in the attached source code fits the ARMA(p,q) model to the time series by using AIC to determine the order of the ARMA model. During the experiments, we also noticed that the best parameter of $(p, q)$ were changing during time, due to the change of  training data. For each month, we use the data in the training window to automatically select the best parameter for the current period. 

\begin{figure}[hbtp]
\centering
\includegraphics[width=7.5cm]{../results/arma_expand_predict.png}
\caption{The Prediction of Returns across Time by ARMA Models using Expanding Window}
\label{fig:arma_expand_predict}
\end{figure}

The prediction of the returns during the test period by the trained model is plotted in Figure. \ref{fig:arma_expand_predict}. The black line in the middle is the predicted returns by the model. The two dash lines on the top and the bottom are the boundaries defined by the standard error. All the real returns are shown as red dots in the chart. We can see most of the returns are falling into the prediction region. The overall precision (defined by the percentage of periods, that fall into the predicted region) are shown in Figure. \ref{fig:arma_expand_prec}. The average performance is about $60\%$. The prediction performance is quite similar to the one using rolling window instead of expansion window. We don't show the charts here due to the limit of the space.

\begin{figure}[hbtp]
\centering
\includegraphics[width=7.5cm]{../results/arma_expand_precision.png}
\caption{The Overall Precision of Prediction of ARMA Models using Expanding Window}
\label{fig:arma_expand_prec}
\end{figure}
